# On the stable recovery of deep structured linear networks under sparsity constraints

arXiv: https://arxiv.org/abs/1706.00342 · HTML: https://mlsanity.com/paper/1706.00342/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 May 2017.

## Built on by (most cited first, 5 of 5)

- [Parameter identifiability of a deep feedforward ReLU neural network](https://mlsanity.com/paper/2112.12982/) (2021)
- [An Embedding of ReLU Networks and an Analysis of their Identifiability](https://mlsanity.com/paper/2107.09370/) (2021)
- [Local Identifiability of Deep ReLU Neural Networks: the Theory](https://mlsanity.com/paper/2206.07424/) (2022)
- [Efficient Identification of Butterfly Sparse Matrix Factorizations](https://mlsanity.com/paper/2110.01230/) (2021)
- [Identifiability in Two-Layer Sparse Matrix Factorization](https://mlsanity.com/paper/2110.01235/) (2021)

## Abstract (arXiv)

We consider a deep structured linear network under sparsity constraints. We study sharp conditions guaranteeing the stability of the optimal parameters defining the network. More precisely, we provide sharp conditions on the network architecture and the sample under which the error on the parameters defining the network scales linearly with the reconstruction error (i.e. the risk). Therefore, under these conditions, the weights obtained with a successful algorithms are well defined and only depend on the architecture of the network and the sample. The features in the latent spaces are stably defined. The stability property is required in order to interpret the features defined in the latent spaces. It can also lead to a guarantee on the statistical risk. This is what motivates this study. The analysis is based on the recently proposed Tensorial Lifting. The particularity of this paper is to consider a sparsity prior. This leads to a better stability constant. As an illustration, we detail the analysis and provide sharp stability guarantees for convolutional linear network under sparsity prior. In this analysis, we distinguish the role of the network architecture and the sample input. This highlights the requirements on the data in connection to parameter stability.